Krzys Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 43, 3/11/6 Author of the Course FM manual available from www.neas-seminars.com/registration Seminar for exam FM: http://www.math.ilstu.edu/actuary/prepcourses.html Society of Actuaries Sample Examination 140-83-94, Problem No. 8 Bill purchases an annuity at a price of 10,000. The annuity makes payments of 500 at the beginning of every 6 months for 20 years. The payments are reinvested in a fund, which earns interest at an annual effective rate i. Interest payments are received every 6 months and reinvested at an nominal rate of 6%, convertible semiannually. Bill realizes an overall effective annual yield of 7% on his original investment over the 20-year period. Calculate i. A. 5.90% B. 6.05% C. 6.20% D. 6.35% E. 6.50% Solution. Since Bill realizes an overall effective annual yield of 7% over 20 years, he ends up with 10,000 !1.0720 " 38,696.84. The pattern of accumulation implies that

10,000 !1.0720= 500 ! 40 + 500 ! 1+ i "1( )( ) ! Is( )40 3% =

= 20,000 + 500 ! 1+ i "1( ) !!!s40 3%

" 40

0.03.

Therefore,

38,696.84 ! 20,000 + 500 " 1+ i #1( ) "!!s40 3%

# 40

0.03!

! 20,000 + 627,721.63 " 1+ i #1( ).

This gives 1+ i !1 " 2.98%

and i ! 1.02982 "1 ! 6.05%. Answer B. Copyright 2006 by Krzysztof Ostaszewski. All rights reserved. Reproduction in whole or in part without express written permission from the author is strictly prohibited. Exercises from the past actuarial examinations are copyrighted by the Society of Actuaries and/or Casualty Actuarial Society and are used here with permission.